3

3. [Maximum mark: 22]

A skip is a container used to carry garbage away from a construction site. For safety reasons the garbage must not extend beyond the top of the skip. The maximum volume of garbage to be removed is therefore equal to the volume of the skip.

A particular design of skip can be modelled as a prism with a trapezoidal cross section. For the skip to be transported, it must have a rectangular base of length 10 m and width 3 m. The length of the sloping edge is fixed at 4 m, and makes an angle $\theta$ with the horizontal.

The following diagram shows such a skip.

(a) Find the volume of this skip,
(i) if the length of the top edge of the skip is 11 m.

(ii) if the height of the skip is 3.2 m.

(iii) if $\theta$ is $\frac{\pi}{3}$. [9]

(b) Show that the volume, $V \text{ m}^3$, of the skip is given by $24 \sin(\theta)(5 + \cos(\theta))$. [2]

(c) Explain, in context, why $\theta \neq 0$. [1]

(d) (i) Sketch the graph of $V = 24 \sin(\theta)(5 + \cos(\theta))$, $0 < \theta < \frac{\pi}{2}$.

(ii) Find the maximum volume of the skip and the value of $\theta$ for which this maximum volume occurs. [4]

(e) Show, by differentiation, that the maximum volume occurs at a value of $\theta$ that satisfies the equation $2 \cos^{2}\theta + 5 \cos\theta - 1 = 0$. [6]